On Generalized Edge Corona Product of Graphs

TL;DR

This paper introduces the generalized edge corona product of graphs, explores its properties related to chromatic number, domination, and independence, and computes these invariants for specific graph classes.

Contribution

It defines the generalized edge corona product and derives new results on its chromatic, domination, and independence numbers for various graph families.

Findings

Computed the k-distance chromatic number for specific graph classes.

Derived formulas for domination and independence numbers under the operation.

Extended understanding of graph invariants in the context of generalized edge corona products.

Abstract

Let $G$ be a simple graph with $m$ edges and $H_i$, $1\leq i \leq m$ be simple graphs too. The generalized edge corona product of graphs $G$ and $H_1, ..., H_m$, denoted by $G \diamond (H_1, ..., H_m)$, is obtained by taking one copy of graphs $G$, $H_1, ..., H_m$ and joining two end vertices of $i$-th edge of $G$ to every vertex of $H_i$, $1\leq i \leq m$. In this paper, some results regarding the $k$-distance chromatic number of Generalized edge corona product of graphs are presented. Also, as a consequence of our results, we compute this invariant for the graphs $K_n \diamond (H_1, ..., H_m)$, $T\diamond (H_1, ..., H_m)$ and $K_{m,n} \diamond (H_1, ..., H_m)$. Moreover, the domination set, domination number and the independence number of any connected graph $G$ and arbitrary graphs $H_i$, $1\leq i \leq |E(G)|$, are evaluated under generalized edge corona operation.